Prior studies on the FL equation mainly concentrate on constructing the soliton solutions and deriving the long time asymptotics. Lenells and Fokas are the first to discover the soliton solutions to the FL equation via the inverse scattering transform and the dressing methods [1]}, [2]}.
Matsuno obtained the bright and the dark soliton solutions to the FL equation via the Hirota method [3]}, [4]}. Later,
Liu et al. derived the double Wronskian solutions to the FL equation via a bilinear approach [5]}, Vekslerchiket et la. constructed the lattice representation and the n-dark solitons of the FL equation in [6]}, while Wright et al. found the breather solutions of the FL equation via a dressing-Bäcklund transformation related to the Riemann-Hilbert (RH) problem formulation [7]}.
A kind of rogue wave solution to the FL equation were obtained by using the Darboux transformation [8]}. The Fokas method was used to investigate the initial-boundary value problem
for the FL equation on the half-line and a finite interval, respectively [9]}, [10]}.
An algebro-geometric method was used to obtain algebro-geometric solutions for the FL equation [11]}. The explicit one-soliton of the initial value problem for the FL equation
was given by the RH method [12]}.
The inverse scattering transform for the FL
equation with nonzero boundary conditions was further investigated by using the RH method [13]}.
Recently the perturbation theory was also used to obtain exact solution to the Fokas-Lenells equation [14]}.
On long time asymptotics, our recent works respectively characterized the long time behaviors of defocusing and focusing FL
equations [15]}, [16]}. However, little is known on the existence of local and global solutions to the FL equation.
In recent years, the global well-posedness of the periodic initial value problem for the FL equation in a Sobolev space was proved by Fokas and Himonas [17]}.
But to the best of our knowledge, the existence of local and global solutions to the FL equation on the line is still unknown.
